Abstract: The interference of two Bose-Einstein condensates, initially in Fock states,can be described in terms of their relative phase, treated as a random unknownvariable. This phase can be understood, either as emerging from themeasurements, or preexisting to them; in the latter case, the originatingstates could be phase states with unknown phases, so that an average over alltheir possible values is taken. Both points of view lead to a description ofprobabilities of results of experiments in terms of a phase angle, which playsthe role of a classical variable. Nevertheless, in some situations, thisdescription is not sufficient: another variable, which we call the -quantumangle-, emerges from the theory. This article studies various manifestations ofthe quantum angle. We first introduce the quantum angle by expressing two Fockstates crossing a beam splitter in terms of phase states, and relate thequantum angle to off-diagonal matrix elements in the phase representation. Thenwe consider an experiment with two beam splitters, where two experimenters makedichotomic measurements with two interferometers and detectors that are farapart; the results lead to violations of the Bell-Clauser-Horne-Shimony-Holtinequality valid for local-realistic theories, including classicaldescriptions of the phase. Finally, we discuss an experiment where particlesfrom each of two sources are either deviated via a beam splitter to a sidecollector or proceed to the point of interference. For a given interferenceresult, we find -population oscillations- in the distributions of the deviatedparticles, which are entirely controlled by the quantum angle. Various versionsof population oscillation experiments are discussed, with two or threeindependent condensates.